aa r X i v : . [ m a t h . C O ] M a y HOM-POLYTOPES
TRISTRAM BOGART ⋆ , MARK CONTOIS, AND JOSEPH GUBELADZE † Abstract.
We study the polytopes of affine maps between two polytopes – the hom-polytopes . The hom-polytope functor has a left adjoint – tensor product poly-topes . The analogy with the category of vector spaces is limited, as we illustrateby a series of explicit examples exhibiting various extremal properties. The mainchallenge for hom-polytopes is to determine their vertices. A polytopal analogue ofthe rank-nullity theorem amounts to understanding how the vertex maps behaverelative to their surjective and injective factors. This leads to interesting classes ofsurjective maps. In the last two sections we focus on two opposite extremal cases– when the source and target polytopes are both polygons and are either genericor regular. introduction Motivation.
The convex polytopes and their affine maps form a natural habi-tat for a major part of the contemporary combinatorics (combinatorial commutativealgebra, toric algebraic geometry, tropical geometry, Ehrhart theory, linear and in-teger programming). The category of polytopes is also an object of study in itsown right. The importance of such an approach is highlighted on the last pagesof [13]. By analogy with algebraic structures and topological spaces, one could askwhether polytopes are also amenable to a unifying analysis, which would providea general context for various important constructions and results. In the case ofalgebra and topology such a unifying machine is homology theory. Since convexpolytopes are just one step away from the classical linear world, the question can beput in very concrete terms: what are the polytopal versions of Hom, ⊗ , Ker, Coker,Ext? How do they fit into the current trends in polytope theory? Is there a moreuniversal (algebraic?) mechanism for addressing concrete challenges in polytopesthan, say, triangulations, analogous to the triangulations vs. homology dichotomyfor topological spaces?A notable example that suggests that these are natural questions is the Billera-Sturmfels concept of fiber polytopes [3]. This construction is expected to be theright kernel object in the category of polytopes. But the analogy with ker is yet tobe fully explained and, more importantly, pushed further to include still conjecturalco-kernel objects in the same category. Mathematics Subject Classification.
Primary 52B11, 52B12; Secondary 5E99, 14-04,18D20, 52B05.
Key words and phrases. polytope, affine map, hom-polytope, polygon, vertex map, category ofpolytopes. ⋆ Supported by NSF grant DMS-0441170. † Supported by NSF grant DMS-1000641.
In this paper we undertake the first step in the direction of categorial analysis ofconvex polytopes: we study the sets of affine maps between two given polytopes, the hom-polytopes . Curiously enough, apart from the motivation above, hom-polytopesshow some relevance in quantum physics [9].1.2.
Results.
That the hom-polytopes are in fact polytopes is a folklore fact; seeSection 2. The full blown analogy with vector spaces is a symmetric closed monoidalstructure on the category of polytopes over which the category itself is enriched(Corollary 3.5). In particular, there is a natural tensor product of polytopes, sat-isfying the usual conjunction with hom-polytopes. But, unlike the linear situation,the tensor product of polytopes exhibits interesting extremal properties (Example3.7). The material up to Corollary 3.5 is modeled on vector spaces and the argu-ments are mostly skipped. The summary is given for the sake of completeness. Afterposting the preprint on arXiv, we learned about Valby’s undergraduate thesis [12]which gives a very detailed treatment of the same material.The description of the facets of hom-polytopes is very simple: a facet consists ofthe maps mapping a chosen vertex of the source into a chosen facet of the target.It is, therefore, the determination of vertices which amounts to understanding thegeometric consequences of our categorial-polytopal endeavor. Very rarely can onehope for a full description of the vertices of hom-polytopes in terms of the sourceand target polytopes. In Section 4 we introduce several tractable classes of surjec-tive vertex maps in arbitrary dimension ( deflations, face collapses ). Their analysis(Theorems 4.1 and 4.2), in particular, yield complete description of the rank 1 ver-tex maps, i. e., the vertex maps whose images in the target polytope are segments(Corollary 4.3).In Section 5 we present several examples of interesting vertex maps, making clear:(i) the limitation of the analogy between the categories of polytopes and vectorspaces ( vertex factorization of non-vertex maps, gaps in ranks ), and (ii) the distinc-tion between the classes of surjective vertex maps, introduced in Section 4.In Section 6 we are able to show that in the hom-polytope between two genericpolygons, all but a few vertices are simple (Theorem 6.1), and we completely de-scribe the exceptions. (To this end we must first introduce an appropriate algebraicparametrization of the set of pairs of polygons.) The result is shown in two steps:(i) introducing combinatorial structures that reduce the problem to the claim thatcertain explicit multivariate polynomial determinants of 31 different types are non-degenerate, and then (ii) verifying the claim by effective methods, with use of thecomputer algebra system
Macaulay 2 .Regular polygons give rise to interesting arithmetic functions – the number of ver-tices of the hom-polytope Hom( P n , P m ) between regular n - and m -gons. In Section7, by explicit polygonal constructions, we obtain the explicit full lists of the verticesof Hom( P m , P n ) when min( m, n ) ≤
4. At the end of Section 7 we present computa-tional results, based on
Polymake , for the number of vertices of Hom( P m , P n ) with m, n ≤ OM-POLYTOPES 3
Affine geometry.
In this and next two subsections we fix terminology andcollect several general facts.For the unexplained background material on convex polyhedral geometry thereader is referred to [4, Ch.1] and [13, Ch.1,2].We will work exclusively in finite dimensional real vector spaces. An affine sub-space of a vector space is the sum of a linear subspace and a vector. For a vectorspace E and a subset X ⊂ E , the affine hull aff( X ) is the minimal affine subspaceof E containing X , and the linear hull lin( X ) is the parallel translate of aff( X )containing 0: aff( X ) = { λ x + · · · + λ n x n | n ∈ N , x , . . . , x n ∈ X,λ , . . . , λ n ∈ R , λ + · · · + λ n = 1 } ;lin( X ) = { µ ( x − x ) + · · · + µ n ( x n − x ) | n ∈ N , x , x , . . . , x n ∈ X,µ , . . . , µ n ∈ R } . An affine map f : E → E ′ between two vector spaces is the composition of alinear map E → E ′ with a parallel translation E ′ → E ′ . A map f : E → E ′ is affineif and only if it respects barycentric coordinates: f n X i =1 λ i x i ! = n X i =1 λ i f ( x i )for all n ∈ N , x , . . . , x n ∈ E , and λ , . . . , λ n ∈ R with P ni =1 λ i = 1. More generally,let A ⊂ E and A ′ ⊂ E ′ be affine subspaces. A map f : A → A ′ is affine if it isa restriction of an affine map between the ambient vector spaces. The set of affinemaps between E and E ′ , denoted by aff( E, E ′ ), is a vector space in a natural way.The set of affine maps A → A ′ , denoted by aff( A, A ′ ), becomes an affine subspace ofaff( E, E ′ ) upon choosing an affine projection π : E → A with π = π and applyingthe embedding: aff( A, A ′ ) → aff( E, E ′ ) , f ι ◦ f ◦ π, where ι : A ′ → E ′ is the identity embedding. The resulting affine structures onaff( A, A ′ ) for various π are all isomorphic. As a result, there is a well-defined notionof convexity in aff( A, A ′ ). This space satisfies dim aff( A, A ′ ) = dim A dim A ′ +dim A ′ .For a subset of an affine space X ⊂ A , its convex hull will be denoted by conv( X ).For X ⊂ E , E a vector space, the conical hull of X will be denoted by cone( X ):cone( X ) = { R + x + · · · + R + x n | n ∈ N , x , . . . , x n ∈ X } , where R + refers to the set of nonnegative reals.For a convex subset X ⊂ A , the relative interior of X in aff( X ) ⊂ A will bedenoted by int( X ). The boundary of X is defined by ∂X = X \ int( X ).For two convex sets X and Y , hom( X, Y ) denotes the set of affine maps X → Y and aff( X, Y ) denotes the set of affine maps aff( X ) → aff( Y ). The natural embed-ding hom( X, Y ) ֒ → aff( X, Y ) makes hom(
X, Y ) into a convex subset of aff(
X, Y )that satisfies:
T. BOGART, M. CONTOIS, AND J. GUBELADZE (1) aff(
X, Y ) = aff(hom(
X, Y )) = hom( X, aff( Y )),(2) int(hom( X, Y )) = hom( X, int( Y )).1.4. Polytopes and cones.
We only consider convex polytopes, i. e., our polytopesare the compact intersections of finitely many affine half-spaces, or equivalently theconvex hulls of finitely many points.For a polytope P , the sets of its facets and vertices will be denoted by F ( P ) andvert( P ), respectively.For two polytopes in their ambient vector spaces P ⊂ E and Q ⊂ E ′ , their join is defined byjoin( P, Q ) = conv { ( x, , , (0 , , y ) | x ∈ P, y ∈ Q } ⊂ E ⊕ R ⊕ E ′ . Let ι P and ι Q be the obvious embeddings of P and Q into join( P, Q ). Every point z ∈ join( P, Q ) has a unique representation z = λι P ( x ) + (1 − λ ) ι Q ( y ), λ ∈ [0 , f : P → R and g : Q → R we have the affine map:join( P, Q ) → R, λι P ( x ) + (1 − λ ) ι Q ( y ) λf ( x )+(1 − λ ) g ( y ) ,λ ∈ [0 , , x ∈ P, y ∈ Q. It is uniquely determined by its restriction to Im ι P and Im ι Q .By cones we will refer to pointed, convex, polyhedral cones: those that are ob-tained as the intersection of finitely many half-spaces and contain no lines. Further notation.
The bipyramid over a polytope P in a vector space E is thepolytope ♦ ( P ) = conv(( P, , c P + (0 , , c P − (0 , ⊂ E ⊕ R , where c p ∈ P is the barycenter.The n -dimensional standard simplex, cube, and cross-polytope are defined by∆ n = conv( e , . . . , e n , e n +1 ) , (cid:3) n = conv (cid:18) n X i =1 δ i e i , δ i = ± (cid:19) , ♦ n = conv( ± e , . . . , ± e n ) , where e i denotes the i th standard basis vector.For a natural number n ≥
3, the standard regular n -gon is P n = conv(1 , ζ n , ζ n , . . . , ζ n − n ) ⊂ C = R ⊕ R ,ζ n = cos(2 π/n ) + sin(2 π/n ) i. Categories.
Our category theory terminology follows the classical source [8]:comma categories, limits and co-limits of diagrams, conjugated functors, and sym-metric monoidal categories. For the concept of enriched categories, we refer thereader to [7]. Even if this is the first time the reader encounters this terminology,the exposition is sufficiently self-explanatory to warrant skipping inclusion of thedefinitions. However, the interested reader can consult [12].
OM-POLYTOPES 5
Let Pol denote the category of polytopes and affine maps and Cones denote thecategory of cones and linear maps. In both categories, for objects A and B we willuse the notation Hom( A, B ) for the corresponding hom-sets.In Pol we have the following universal equalities:join(
P, Q ) = P a Q = lim → ( P, Q ) ,P × Q = P Y Q = lim ← (cid:0) P, Q (cid:1) . Hom-polytopes
The following proposition is well-known [13, § Polymake [1, 5] for computing hom-polytopes. We used this software forthe experiments presented in Section 7 below. The details are written up to easethe references in the following sections.
Proposition 2.1.
Let
P, Q, R be polytopes. (1) Hom(
P, Q ) is a polytope in aff( P, Q ) with F (Hom( P, Q )) = (cid:8) H ( v, F ) | v ∈ vert( P ) , F ∈ F ( Q ) ,H ( v, F ) = { f ∈ Hom(
P, Q ) | f ( v ) ∈ F } (cid:9) , (2) dim Hom( P, Q ) = dim P dim Q + dim Q . (3) Hom( P, Q × R ) ∼ = Hom( P, Q ) × Hom(
P, R ) , (4) Hom( P, Q ∩ R ) ∼ = Hom( P, Q ) ∩ Hom(
P, R )(5) Hom(join(
P, Q ) , R ) ∼ = Hom( P, R ) × Hom(
Q, R ) .Proof. (1) For every facet F ⊂ Q we fix a surjective affine map ϕ F : Q → R + ,vanishing only on F . Claim . The system of affine maps ϕ F,v : aff(
P, Q ) → R , f ( ϕ F ◦ f )( v ) , F ∈ F ( Q ) , v ∈ vert( P ) , defines the facets of Hom( P, Q ).The equality Hom(
P, Q ) = \ F,v { f ∈ aff( P, Q ) | ϕ F,v ( f ) ≥ } is straightforward. But for any vertex v ∈ P and any facet F ⊂ Q , there is an affinemap f : P → Q such that f ( v ) ∈ int( F ) and f ( w ) ∈ int( Q ) for each vertex w ∈ P with w = v . To obtain such map, take the composition of a parallel projectionof P onto [0 , v to 0 and the rest of P onto (0 , , → Q , mapping 0 to int( F ) and (0 ,
1] to int( Q ). Thus, f ∈ ϕ F,v (0) − \ [ F ( Q ) \ { F } w ∈ vert( P ) \ { v } ϕ G,w (0) − , Called mapping polytopes in Polymake . T. BOGART, M. CONTOIS, AND J. GUBELADZE which proves the claim.(2) In the notation introduced above, we haveint(Hom(
P, Q )) = \ F,v { f ∈ aff( P, Q ) | ϕ F,v ( f ) > } , a nonempty bounded open subset of aff( P, Q ) defined by linear inequalities, soHom(
P, Q ) ⊂ aff( P, Q ) is a full-dimensional polytope.The parts (3,4,5) follow from the universal equalities in Section 1.4 and the naturalbijections of sets for any object a and any diagram D in any category:Hom( a, lim ← D ) ∼ = lim ← Hom( a, D ) , Hom(lim → D , a ) ∼ = lim ← Hom( D , a );these bijections are affine maps in our polytopal setting. (cid:3) For cones we have the following analogous statment:
Proposition 2.2.
Let C , C , C be cones. (1) Hom( C , C ) is a cone in Hom(lin C , lin C ) , whose facets are naturally in-dexed by the pairs ( R, F ) , R ⊂ C an extremal ray and F ⊂ C a facet. (2) Hom( C , C × C ) ∼ = Hom( C , C ) × Hom( C , C ) . (3) Hom( C × C , C ) ∼ = Hom( C , C ) × Hom( C , C ) . (4) Hom( C , C ∩ C ) ∼ = Hom( C , C ) ∩ Hom( C , C ) . Homogenization, duals, and tensor product
To a polytope P in a vector space E we associate the homogenization cone : C ( P ) = cone( P, , where ( P,
1) = { ( x, | x ∈ P } ⊂ E ⊕ R . A graded cone is a cone C together with an affine map ϕ : C → R + , satisfyingthe condition ϕ − (0) = { } . The map ϕ is called a grading . All of our cones admita grading [4, Prop.1.21]. For a graded cone ( C, ϕ ) we have the de-homogenizationpolytope ϕ − (1), denoted by C [1] .For a full-dimensional polytope P with 0 ∈ int( P ) and a full dimensional cone C in a vector space E , the corresponding dual objects in the dual space E ø are definedby P ø = { h ∈ E ø | h ( x ) ≤ x ∈ P } ,C ø = { h ∈ E ø | h ( C ) ⊂ R + } = Hom( C, R + ) . The duals are also full-dimensional in their ambient space E ø .For the general facts of dual polytopes and dual cones the reader is referred to [4, § § P in a vector space E , we consider thecone C ( P ) as a graded cone w.r.t. the grading C ( P ) E ⊕ R pr R R . OM-POLYTOPES 7
For two polytopes P and Q , let ϕ and ψ denote the gradings C ( P ) , C ( Q ) → R + ,respectively. Then we have the grading on the product of the homogenization cones: C ( P ) × C ( Q ) → R + , ( x, y ) → ϕ ( x ) + ψ ( y ) , which results in the isomorphism of polytopes( C ( P ) × C ( Q )) [1] ∼ = join( P, Q ) . Further, if dim P = dim E and 0 ∈ int( P ) (i. e., (0 , ∈ int( P, C ( P ) ø will be viewed as a graded cone via the grading C ( P ) ø → R + , h h ((0 , h ((0 , h = 0. Definition 3.1.
Let E and E ′ be vector spaces, C ⊂ E and C ′ ⊂ E ′ be cones, and P ⊂ E and Q ⊂ E ′ be polytopes. We define the tensor products as follows: C ⊗ C ′ = cone { ( x ⊗ y ) | x ∈ C, y ∈ C ′ } ⊂ E ⊗ E ′ ,P ⊗ Q = conv (cid:0)(cid:8) ( x ⊗ y, x, y ) | x ∈ P, y ∈ Q (cid:9)(cid:1) ⊂ (cid:0) E ⊗ E ′ (cid:1) ⊕ E ⊕ E ′ . Proposition 3.2.
Let
C, C , C , C be cones and P be a polytope. (1) The bilinear map C × C −→ C ⊗ C , ( x, y ) x ⊗ y , solves the followinguniversal problem: any bilinear map C × C −→ C passes through a uniquelinear map ϕ , C × C f C ⊗ C (cid:8) ∃ ! ϕ C . Equivalently,
Hom( C ⊗ C , C ) ∼ = Hom( C , Hom( C , C )) , i. e., ⊗ , Hom :Cones × Cones → Cones form a pair of left and right adjoint functors. (2) Hom( C , C ø2 ) ø ∼ = C ⊗ C ∼ = Hom( C , C ø1 ) ø for C and C full-dimensional. (3) dim( C ⊗ C ) = dim C dim C = dim Hom( C , C ) ,C ⊗ R + ∼ = C, ( C ⊗ C ) ⊗ C ∼ = C ⊗ ( C ⊗ C ) , C ⊗ C ∼ = C ⊗ C ,C ⊗ ( C × C ) ∼ = ( C ⊗ C ) × ( C ⊗ C ) . (4) The extremal rays of C ⊗ C are the tensor products of the extremal rays of C and C . (5) Hom( P, C ) ∼ = Hom( C ( P ) , C ) , where the Hom on the right refers to the setof affine maps from P to C . (6) If P is full-dimensional and ∈ int( P ) then: C ( P ø ) ∼ = C ( P ) ø as graded cones, i. e., P ø ∼ = C ( P ) ø[1] , and Hom( P, [0 , ∼ = C ( P ø ) ∩ ((0 , − C ( P ø )) . Proof. (1–3) are straightforward analogues of the corresponding linear algebra facts.
Notice.
Actually, (2) holds true for general, not necessarily full dimensional, cones.This is so because C øø = C for general cones. Here we have to restrict to the T. BOGART, M. CONTOIS, AND J. GUBELADZE full dimensional case because from the beginning we have restricted to the case ofpointed cones, and the dual of a cone is pointed iff the cone is full dimensional.(4) Let R i ⊂ C i be extremal rays, ν i : C i → R + be linear maps with ν − i (0) = R i ,and ϕ i : C i → R + be gradings, i = 1 ,
2. Then the linear map ν ⊗ ϕ + ϕ ⊗ ν : C ⊗ C → R + satisfies the condition ( ν ⊗ ϕ + ϕ ⊗ ν ) − (0) = R ⊗ R . Conversely, let ξ = P kj =1 x j ⊗ y j be an element of an extremal ray R ⊂ C ⊗ C ,where x j ∈ C \ { } and y j ∈ C \ { } for all j . Then x j ⊗ y j ∈ R for all j because x j ⊗ y j = 0 for all j – the tensor product of two nonzero vectors is a nonzero vector.Consequently, there exist real numbers t j > x j ⊗ y j = ( t j x ) ⊗ y for every j .In particular, R = R + ( x ⊗ y ). All we need to show is that x and y are extremalgenerators of C and C , respectively. Without loss of generality we can assumethat x is not an extremal generator of C . There is a segment [ u, v ] ∈ C such that x ∈ ( u, v ) and 0 / ∈ aff([ u, v ]). But then the subset [ u ⊗ y , v ⊗ y ] ⊂ C ⊗ C is anondegenerate segment and 0 / ∈ aff( u ⊗ y , v ⊗ y ) – one uses the same fact on thetensor product of nonzero vectors. This contradicts the assumption that x ⊗ y ∈ R .(5) We have the mutually inverse linear maps: α : Hom( P, C ) → Hom( C ( P ) , C ) , α ( h )(( x, z )) = h ( z − x ) , ( x, z ) ∈ C ( P ) \ { } ,β : Hom( C ( P ) , C ) → Hom(
P, C ) , β ( h )( x ) = h (( x, , x ∈ P. (6) We have the following mutually inverse affine maps: γ : P ø → C ( P ) ø[1] , γ ( h )(( x, z )) = z − z · h ( z − x ) , ( x, z ) ∈ C ( P ) \ { } ,δ : C ( P ) ø[1] → P ø , δ ( h )( x ) = 1 − h (( x, , x ∈ P. That γ and δ in fact evaluate in the right objects and have all necessary propertiesis straightorward.For the second isomorphism, we note that [0 ,
1] = R + ∩ (1 − R + ) and so (5), theprevious isomorphism, and Proposition 2.2(4) apply. (cid:3) Remark 3.3. (1) Caution is needed in extending linear algebra facts to cones. Forinstance, the canonical map C ø1 ⊗ C ø2 → ( C ⊗ C ) ø is in general not an isomorphism:by Proposition 3.2(4), the extremal rays of the first cone naturally correspond to thepairs of facets F ⊂ C and F ⊂ C , while those of the second one are in bijectivecorrespondence with the facets of C ⊗ C . But there is no natural correspondencebetween the two sets. In fact, they can easily be non-bijective – a combined effectof Proposition 2.1(1) and the upcoming Proposition 3.4(1).(2) As a consequence of (6), for two isomorphic full dimensional polytopes P ∼ = P with 0 ∈ int( P ) ∩ int( P ), although P ø1 and P ø2 are in general not isomorphic in Pol,we nonetheless have the isomorphism C ( P ø1 ) ∩ ((0 , − C ( P ø1 )) ∼ = C ( P ø2 ) ∩ ((0 , − C ( P ø2 )) . OM-POLYTOPES 9
In particular, we recover the standard fact that, when a full-dimensional polytopemoves around the origin so that the origin stays in the interior of the polytope, theresulting dual polytopes are all projectively equivalent.For polytopes
P, Q, R , a map P × Q → R is called bi-affine if, upon fixing onecomponent, it is affine w.r.t. the other component. Proposition 3.4.
Let
P, Q, R be polytopes. (1) C ( P ⊗ Q ) ∼ = C ( P ) ⊗ C ( Q ) , or, equivalently, vert( P ⊗ Q ) = { ( v ⊗ w, v, w ) | v ∈ vert( P ) , w ∈ vert( Q ) } . (2) The bi-affine map P × Q −→ P ⊗ Q , ( x, y ) ( x ⊗ y, x, y ) , solves the followinguniversal problem: any bi-affine map P × Q −→ R passes through a uniqueaffine map ϕ , P × Q f P ⊗ Q (cid:8) ∃ ! ϕ R .
Equivalently,
Hom( P ⊗ Q, R ) ∼ = Hom( P, Hom(
Q, R )) , i.e., ⊗ , Hom : Pol × Pol → Pol form a pair of left and right adjoint functors. (3) dim( P ⊗ Q ) = dim P dim Q + dim P + dim Q,P ⊗ {∗} ∼ = P, ( P ⊗ Q ) ⊗ R ∼ = P ⊗ ( Q ⊗ R ) , P ⊗ Q ∼ = Q ⊗ P.P ⊗ join( Q, R ) ∼ = join( P ⊗ Q, P ⊗ R ) , vert( P ⊗ Q ) = { ( v ⊗ w, v, w ) | v ∈ vert( P ) , w ∈ vert( Q ) } . Proof. (1) We have C ( P ) ⊗ C ( Q ) =cone { ( x ⊗ y ) | x ∈ C ( P ) , y ∈ C ( Q ) } =cone { ( x ⊗ y ) | x ∈ ( P, , y ∈ ( Q, } =cone { ( u ⊗ v, u, v, | u ∈ P, v ∈ Q } = C ( P ⊗ Q ) . (2) A bi-affine map ψ : P × Q R gives rise to the following bilinear map: C ( P ) × C ( Q ) C ( R ) , (( u, x ) , ( v, y )) (cid:0) xy · ψ (cid:0) x − u, y − v (cid:1) , xy (cid:1) , x, y > . By Proposition 3.2(1), we have the linear map: C ( P ) ⊗ C ( Q ) → C ( R ) , ( u, x ) ⊗ ( v, y ) (cid:0) xy · ψ (cid:0) x − u, y − v (cid:1) , xy (cid:1) , x, y > , which shows that the composite affine map P ⊗ Q ∼ = ( P ⊗ Q, ֒ → C ( P ⊗ Q ) ∼ = C ( P ) ⊗ C ( Q ) → C ( R ) , evaluates in ( R, (cid:3) For polytopes
P, Q, R , the pairingHom(
P, Q ) × Hom(
Q, R ) → Hom(
P, R ) , ( f, g ) g ◦ f, is clearly bi-affine. The pentagon and hexagon coherence conditions of the bifunctor ⊗ : Pol × Pol → Pol are inherited from the corresponding conditions of the tensorproduct of vector spaces. So Proposition 3.4 has the following consequence.
Corollary 3.5. (Pol , ⊗ ) is a symmetric closed monoidal category w.r.t. which Pol is self-enriched.
As a more computational application, we have the following.
Corollary 3.6.
Let P be a polytope and n and m be natural numbers. Then: (1) Hom(∆ n , P ) ∼ = P n +1 and ∆ n ⊗ P ∼ = P join n +1 – the ( n + 1) -fold iteration ofjoin, applied to P . (2) Hom( P, (cid:3) n ) ∼ = ♦ ( P ø ) n for P centrally symmetric w.r.t. . In particular, Hom( (cid:3) m , (cid:3) n ) ∼ = ( ♦ m +1 ) n . (3) Hom( (cid:3) m , ♦ n ) ∼ = Hom( (cid:3) n − , ♦ m +1 ) .Proof. (1) Any map vert(∆ n ) → P extends uniquely to an affine map ∆ n → P .For the second isomorphism, one uses Proposition 3.4(3), the third isomorphism,and the fact that ∆ n ∼ = ( ∗ ) join n +1 .(2) This follows from Propositions 2.1(3) and 3.2(6), the 2nd isomorphism, and thefact that P ø is also symmetric w.r.t. the origin. (See Remark 3.3(2).)(3) One applies the conjunction ⊗ ⊣ Hom twice. (cid:3)
We have not been able to find rank ≥ (cid:3) m , ∆ n )). Moregenerally, one would wish to have Corollary 3.6 completed by an explicit descriptionof the hom-polytopes between the general regular polytopes. Dimensions 3 and 4seem more challenging than dim ≥
5. Currently, even the 2-dimensional case, orjust a satisfactory description of vert(Hom( P n , P m )), is out of reach. Partial resultsin the latter direction are presented in Section 7.In the context of extremal maps from regular polytopes, it is interesting to remarkthat the octahedron ♦ admits an affine embedding into any simple 3-dimensionalpolytope P so that the vertices of ♦ map to the boundary ∂P [2].We conclude the section with one application of the tensor product, yielding pointconfigurations with interesting extremal properties. Example 3.7.
Let P and Q be polytopes in a vector space E .(1) For a vertex v ∈ P and an edge [ w , w ] ⊂ Q , the segment[( v ⊗ w , v, w ) , ( v ⊗ w , v, w )] ⊂ E ⊗ ⊕ E ⊕ is easily seen to be an edge of P ⊗ Q .(2) The tensor square P ⊗ = P ⊗ P admits less obvious edges:[( v ⊗ v, v, v ) , ( w ⊗ w, w, w )] ⊂ P ⊗ , v, w ∈ vert( P ) , v = w. OM-POLYTOPES 11
To show this, assume v, w ∈ vert( P ), v = w and let ϕ v , ϕ w : C ( P ) → R + be linearmaps, vanishing on exactly cone( v ′ ) and cone( w ′ ), respectively, where v ′ = ( v,
1) and w ′ = ( w, C ( P ⊗ ) ∼ = C ( P ) ⊗ ϕ v ⊗ ϕ w + ϕ w ⊗ ϕ v R + (the isomorphism on the left is from Proposition 3.4(1)), vanishes on exactly the2-face cone( v ′ ⊗ v ′ , w ′ ⊗ w ′ ) ⊂ C ( P ⊗ ) . As a consequence of the two types of edges, if P has m vertices and n edges, thenthe tensor square P ⊗ has at least 2 mn + m ( m − edges. This counting is, however,far from the complete list; an example is (∆ n ) ⊗ = ∆ n +2 n .(3) It follows from (2) that the following polytope is neighborly :conv (cid:0) ( v ⊗ v, v, v ) | v ∈ vert( P ) (cid:1) ⊂ E ⊗ ⊕ E ⊕ , i. e., any two vertices are joined by an edge. (In [13, Ch.0] the property is called .)(4) A similar argument implies that for the unit Euclidean ball B d ⊂ R d and anytwo points v, w ∈ S d − = ∂B d , the segment[( v ⊗ v, v, v ) , ( w ⊗ w, w, w )] ⊂ ( B d ) ⊗ (= conv(( x ⊗ y, x, y ) | x, y ∈ B d ))is an extremal subset ([4, p.10]).(5) Similarly, the map ψ d − : S d − → R d +2 d , v ( v ⊗ v, v, v ) , has the following extremal property: for any system of points v , . . . , v n ∈ S d − , their ψ d − -images are in convex and neighborly position. This observation is, in a sense,weaker than (4) because conv(Im ψ d − ) $ ( B d ) ⊗ . (For instance, ( v ⊗ w, v, w ) ∈ ( B d ) ⊗ \ conv(Im ψ d − ) for v, w ∈ S d − , v = w .) On the other hand, it yields theinteresting embedding ψ : S → R , (cos t, sin t ) (cos t, sin t, cos 2 t, sin 2 t ) , which maps any number of points on the unit circle into a convex neighborly pointconfiguration in R . In fact, ψ is obtained from ψ by the following series of affinetransformations of R (cos t, sin t ) (cos t, sin t, cos t sin t, cos t sin t, cos t, sin t, cos t, sin t ) (cos t, sin t, cos t, sin t, cos t sin t ) (cos t, sin t, cos t, − cos t, sin 2 t ) (cos t, sin t, cos t, sin 2 t ) (cos t, sin t, cos 2 t, sin 2 t ) , implying an affine isomorphism conv(Im ψ ) ∼ = conv(Im ψ ). Vertex factorizations
Let A and B be convex sets. A family of maps (cid:0) f t (cid:1) ( − , ⊂ Hom(
A, B ) is calledan affine 1-family if the map( − , → Hom(
A, B ) , t f t , is injective and affine.A map f : P → Q in Pol is called a vertex map , or just a vertex , if f ∈ vert(Hom( P, Q )).We will need the following obvious (affine) perturbation criteria for vertices andinterior points. Let f : P → Q be an affine map between two polytopes. Then:(pc ) f is not a vertex if and only if there is an affine 1-family (cid:0) f t (cid:1) ( − , ⊂ Hom(
P, Q ), with f = f .(pc ) f ∈ int(Hom( P, Q )) if and only if for any affine 1-family (cid:0) f t (cid:1) ( − , ⊂ aff( P, Q )with f = f there exists a real number ε > (cid:0) f t (cid:1) ( − ε,ε ) ⊂ Hom(
P, Q )Every map f in Pol factors, uniquely up to the obvious equivalence, into a sur-jective and an injective map in Pol: f = f inj ◦ f surj .A map f : P → Q is called a deflation if it satisfies the conditions:(i) f is surjective,(ii) f is a vertex,(iii) for any vertex v ∈ P either f ( v ) ∈ vert( Q ) or f ( v ) ∈ int( Q ).Simple examples of deflations are provided by the parallel projections (cid:3) n → (cid:3) m along the subspace R + e m +1 + · · · + R + e n ⊂ R n , where we assume m < n .The rank of a map f in Pol is defined by rank f = dim Im( f ). Theorem 4.1.
Let f : P → Q be a map in Pol . (1) If f is a vertex then so are f inj and f surj . (2) If vert( f ( P )) ⊂ vert( Q ) and f surj is a deflation then f is also a vertex map. (3) There are examples of non-vertex maps f with f inj a vertex map and f surj adeflation. In particular, the converse to the implication in (1) is not true.Proof. We can assume f surj : P → f ( P ).(1) (Using (pc ).) If f inj : f ( P ) → Q is not a vertex then there exists an affine 1-family ( ψ t ) ( − , ⊂ Hom( f ( P ) , Q ) with ψ = f inj . Then the resulting affine 1-family (cid:0) ψ t ◦ f surj (cid:1) ( − , ⊂ Hom(
P, Q ) has ψ ◦ f surj = f , a contradiction.If f surj is not vertex then there is an affine 1-family ( ϕ t ) ( − , ⊂ Hom(
P, f ( P ))with ϕ = f surj . So we get the affine 1-family (cid:0) f inj ◦ ϕ t (cid:1) ( − , ⊂ Hom(
P, Q ) with f inj ◦ ϕ = f , again a contradiction.(2) Observe that, in view of (pc ), the condition vert( f ( P )) ⊂ vert( Q ) alreadyimplies that f inj is a vertex map. But we do not use this explicitly.Assume to the contrary that f is not a vertex. By (pc ), there is an affine 1-family( f t ) ( − , ⊂ Hom(
P, Q ) with f = f .We will show that the maps ϕ v : ( − , → Q, t f t ( v ) , OM-POLYTOPES 13 are constant for all v ∈ vert( P ). Since this implies that the map( − , → Hom(
P, Q ) , t f t , is constant, we have the desired contradiction.The maps ϕ v : ( − , → Q, t f t ( v ) , are constant for the vertices v ∈ vert( P ) with f ( v ) ∈ vert( Q ).Assume v , . . . , v k are the vertices of P for which the maps ϕ v , . . . , ϕ v k : ( − , → Q, t f t ( v k ) , are not constant. By the hypothesis, f ( v ) , . . . , f ( v k ) ∈ int( f ( P )).Fix a (dim Q − rank f )-dimensional affine subspace H ⊂ aff( Q ) that is parallel tonone of the intervals [ t ∈ ( − , f t ( v i ) ⊂ Q, i = 1 , . . . , k, and consider the parallel projection π : Q → aff(Im( f )) along H . We get an affinefamily ( π ◦ f t ) ( − , ⊂ = aff( P, Im( f )) with π ◦ f = f surj . By (pc ), there exists areal number ε > π ◦ f t ) ( − ε,ε ) ⊂ Hom( P, Im( f )),contradicting the assumption that f surj is vertex.(3) Examples of non-vertex maps f for which f inj is a vertex and f surj is a deflationwill be presented in Example 5.1. (cid:3) Let P be a polytope and 0 ≤ r < dim P . Consider a family Γ = { G , . . . , G k } ,satisfying the conditions:(i) G i ⊂ P is a face and dim G i > i = 1 , . . . , k ,(ii) codim(lin( G ) + · · · + lin( G k )) = r ,(iii) ( G i + lin( G ) + · · · + lin( G k )) ∩ P = G i , i = 1 , . . . , k ,(iv) the family of faces { G , . . . , G k } is maximal w.r.t inclusion among the familiessatisfying the first three conditions.The linear hulls above are taken in the ambient vector space and the codimensionis understood relative to lin( P ).To Γ we associate a surjective affine map of rank r from P as follows. We canassume 0 ∈ P . Consider a linear map f : lin( P ) → E with dim E = r andker( f ) = lin( G ) + · · · + lin( G k ). For the restriction f | P we will use the notation f Γ .A map P → Q in Pol is called a face-collapse of rank r if it is of the type f Γ upto isomorphism in the comma category P ↓ Pol.The following example shows that neither of the conditions (iii) and (iv) abovecan be dropped in the definition of face-collapses. Let P be a regular hexagon withvertices v , . . . , v . Then the pair of facets Γ = { [ v , v ] × [0 , , [ v , v ] × [0 , } of theunit prism P × [0 ,
1] satisfies (i-iv). On the one hand, if one drops (iii), then Γ ceasesto be maximal. On the other hand, if one drops (iv), then Γ ′ = { [ v , v ] × [0 , } satisfies (i,ii,iii), but one cannot collapse the facet [ v , v ] × [0 ,
1] into a point by anaffine map without also collapsing the facet [ v , v ] × [0 , v v v v v v Figure 1.
A polytope illustrating the definition of face-collapse
Theorem 4.2.
We have the following proper containments:Deflations $ Face-Collapses $ Surjective Vertex Maps.Proof.
First we show that face-collapses are vertex maps.Let P ∈ Pol and Γ = { G , . . . , G k } be a system of faces of P satisfying thecondition (i–iv) above. Without loss of generality we can assume 0 ∈ P . Denote H = lin( G ) + · · · + lin( G k ).First we observe that f ( G i ) ∈ vert( f ( P )) for every i = 1 , . . . , k . In fact, f ( G i ) ⊂ ∂f ( P ) by (iii) and dim f ( G i ) = 0. But then, if f ( G i ) ∈ int( γ ) for some positivedimensional face γ ⊂ f ( P ), we have int( G i ) ⊂ int (cid:0) f − ( γ ) (cid:1) and f − ( γ ) is a face of P with G i $ f − ( γ ). This is impossible because G i ⊂ P is a face.Suppose f Γ is not a vertex. By (pc ), there exists an affine 1-family ( f t ) ( − , ⊂ Hom(
P, f ( P )) with f = f Γ . The condition f ( G i ) ∈ vert( f ( P )) forces f t | G i = f | G i for all t ∈ ( − ,
1) and i = 1 , . . . , k . In other words, the 1-family ( f t ) ( − , isconstant on every G i . Let { w k +1 , . . . , w k + l } = vert( f ( P )) \ { f ( G ) , . . . , f ( G k ) } . Fix v k +1 , . . . , v k + l ∈ vert( P ) with f ( v j ) = w j , j = k + 1 , . . . , k + l . (The v j are uniquelydetermined, but we do not need this.) Using again (pc ), the 1-family ( f t ) ( − , isconstant on the v j . So ( f t ) ( − , is constant on conv( G , . . . , G k , v k +1 , . . . , v k + l ). But,by (ii), the latter is a full dimensional subpolytope of P and that forces our 1-familyto be constant on P – a contradiction.Next we show that deflations are face-collapses.Consider a rank r deflation f ∈ vert (cid:0) Hom(
P, f ( P ) (cid:1) . We can assume that 0 ∈ P and that f (0) = 0; i.e., f is the restriction of a (unique) linear map h : lin( P ) → lin( f ( P )). Let w , . . . , w k ∈ f ( P ) be the vertices with dim f − ( w i ) > i = 1 , . . . , k . It is enough to show the following Claim.
The faces G i = f − ( w i ) ⊂ P , i = 1 , . . . , k, form a system satisfying theconditions (i–iv) with ker( h ) = lin( G ) + · · · + lin( G k ). OM-POLYTOPES 15
The conditions (i,iii,iv) are straightforward. Since lin( G )+ · · · +lin( G k ) ⊂ ker( h ),we only need to verify (ii).Assume to the contrary codim (cid:0) lin( G )+ · · · +lin( G k ) (cid:1) > r . Let { w k +1 , . . . , w k + l } =vert( f ( P )) \ { w , . . . , w k } . There are (uniquely determined) vertices v k +1 , . . . , v k + l ∈ P with f ( v j ) = w j for j = k + 1 , . . . , k + l . Our assumption impliesdim conv( G , . . . , G k , v k +1 , . . . , v k + l ) < dim P. Without loss of generality we can additionally assume0 ∈ conv( G , . . . , G k , v k +1 , . . . , v k + l ) . Pick a basis B ⊂ lin( P ), restricting to a basis B ⊂ lin( G , . . . , G k , v k +1 , . . . , v k + l ),and a basis B ′ ⊂ lin( f ( P )). Let M be the matrix of h with respect to the bases B and B ′ so that the first B ) rows correspond to the elements of B . Considerthe affine 1-family ( h t ) ( − , ⊂ Hom(lin( P ) , lin( f ( P ))), where the matrix of h t inthe bases B and B ′ is obtained from M by adding ( t, . . . , t ) ∈ R B ′ to each ofthe last B \ B ) rows. Then, because f is a deflation, there exists 0 < ε < h t ( P ) ⊂ f ( P ) for all − ε < t < ε and this produces an affine 1-family inHom( P, f ( P )), containing f – a contradiction by (pc ).Examples 5.2 and 5.3 in the next section will show that both inclusions in Theorem4.2 are proper inclusions. (cid:3) Corollary 4.3. (1)
A rank surjective map is a vertex if and only if it is aface-collapse if and only if it is a deflation. (2) Every system Γ of faces of P satisfying (i-iv) and rank( f Γ ) = 1 has at mosttwo elements. If P is centrally symmetric then such systems Γ are exactlythe pairs of opposite facets of P . (3) A rank one map f is a vertex iff f surj and f inj are vertices. (4) For a polygon P with l edges, of which m pairs are parallel, and a polytope Q with n vertices, the number of rank vertex maps P → Q is ( l − m ) n ( n − .Proof. (1) holds true because of the first inclusion in Theorem 4.2 and the fact thatthe boundary of a segment is just the vertices of the segment .(2) If { u, v } = vert f Γ ( P ) and dim f − ( u ) , dim − ( v ) > { f − ( u ) , f − ( v ) } ;otherwise Γ consists of a single facet of P . The conclusion for centrally symmetricpolytopes is obvious.(3) From (pc ), the injective vertex maps from a segment to Q are the maps thatsend the two vertices of the segment to distinct vertices of Q . So the claim followsfrom (1) and Theorem 4.1(1,2).(4) In view of (1) and (2), ( l − m ) is the number of surjective vertex maps from P to a segment. The count follows from (3) and the observation above on injectivevertex maps from a segment. (cid:3) v v v v v v w w w w w w H Figure 2.
The construction of Example 5.1 with n = 35. Examples of vertex maps
In this section we present various constructions of vertex maps exhibiting inter-esting phenomena, some of them mentioned in the previous sections.
Example 5.1 ( Vertex factorization of non-vertex maps ) . For a map f : P → Q inPol, if 1 + dim P affinely independent vertices of P map to vertices of Q then f is avertex. In fact, if there were an affine 1-family ( f t ) ( − , ⊂ Hom(
P, Q ) with f = f ,then the family would be constant on the distinguished vertices of P . The affinehull of these vertices is all of aff( P ). Consequently, ( f t ) ( − , must be constant on P – a contradiction. This simple observation, together with Theorem 4.1(2), suggeststhat to look for non-vertex maps f : P → Q with f surj and f inj both vertices, weshould consider maps that keep the vertices of Im f away from vert( Q ). We firstgive a general construction and then construct examples of vertex factorizations ofnon-vertex maps.Let Q ⊂ R be a 3-polytope and H ⊂ R be an affine plane such that: (i) theintersection P = Q ∩ H is a polygon with 2 n ≥ w , . . . , w n ∈ vert( Q ) such that every open interval ( w i , w i +1 ) contains exactly onevertex v i ∈ vert( P ), the indexing being mod 2 n . Observe that the segments [ w i , w i +1 ]are necessarily edges of Q . Claim . The identity embedding ι : P → Q is a vertex of Hom( P, Q ).The two groups of vertices w , w , . . . , w n − and w , w , . . . , w n are separated by H . To see that ι is a vertex, we apply the following sliding argument . Assume ι / ∈ vert(Hom( P, Q )). Pick a vertex of σ ∈ Hom(
P, Q ), which belongs to the sameminimal face of Hom(
P, Q ) as ι . By sliding ι along the segment [ ι, σ ] one gets a1-parameter family ( ι t ) [0 , ⊂ Hom(
P, Q ) with ι = ι and ι = σ such that, forevery t ∈ [0 , ι t is a polygon, isomorphic to P and with vertices OM-POLYTOPES 17 H Figure 3.
The construction of Example 5.1 with P n regular, n = 4 v ti ∈ [ w i , w i +1 ]. In fact, the ‘vertex of ι t ( P ) ↔ edge of Q ’ incidence table remainsconstant in the process of sliding for t ∈ [0 , ι t ) = 2 for t ∈ [0 , σ = 2 as well because no affine line can intersect all segments [ w i , w i +1 ]simultaneously – this is where we use the inequality n ≥ In particular, Im( ι t ) ∼ = P for all t ∈ [0 , t = 1 and σ is a vertex, σ belongs to more facets of Hom( P, Q ) than ι . So the condition v i ∈ [ w i , w i +1 ] for i = 1 , . . . , n forces v j = w j or v j = w j +1 for some j . But thenone easily deduces w , . . . , w n ∈ Im σ , forcing dim(Im σ ) = 3. This contradicts thefact that Im σ ∼ = P , proving the claim.Consider the special case of the construction above, using the regular n -gon P n with n even: Q = conv( ˆ P n , ˇ P n ) ⊂ C ⊕ R , withˆ P n = ( P n , − ⊂ C ⊕ R , ˇ P n = ( ηP n , ⊂ C ⊕ R ,η = cos( π/n ) + sin( π/n ) i. For every h ∈ ( − , P h = Q ∩ ( C , h ), is a centrally symmetric n -gonand the identity embedding ι h : P h → Q is a vertex of Hom( P h , Q ). We can choosesurjective maps ρ h : ♦ n → P h , h ∈ ( − ,
1) so that (i) for every h , vertices map tovertices and (ii) the maps ρ h , viewed as elements of aff( R n , C ⊕ R ), continuouslydepend on h . Because each ρ h maps vertices to vertices, we have ρ h ∈ vert( ♦ n , P h )for every h . Moreover, ρ h is a deflation for every h .It is important that the assignment h ρ h is not just continuous, but even anaffine map ( − , → aff( R n , C ⊕ R ). For n = 2 the construction does not go through; an example is a tetrahedron and a plane,intersecting the tetrahedron in a parallelogram – the identity embedding of the parallelogram intothe tetrahedron is not a vertex. e e e e e π Figure 4.
A face-collapse that is not a deflationSummarizing, for every h ∈ ( − , f h = ι h ◦ ρ h has thesurjective factor ( f h ) surj = ρ h a deflation and the injective factor ( f h ) inj = ι h avertex map. Yet, f h / ∈ vert(Hom( ♦ n , Q )) . For simplicity of notation we can restrict to the case h = 0, and then f fits into theaffine 1-family ( f h ) h ∈ ( − , ⊂ Hom( ♦ n , Q ).From the construction above one can derive new examples of vertex factorizationsof non-vertex maps as follows. Observe that, for every h ∈ ( − , τ h : (cid:3) n → P h . So the composite maps g h = ι h ◦ τ h do not belong tovert(Hom( (cid:3) n , Q )), yet ( g h ) surj is a deflation and ( g h ) inj is a vertex map. Example 5.2 ( Face-collapses that are not deflations ) . Not all face-collapses aredeflations. Here is an example. Let { e , e , e } ⊂ R be the standard basis andconsider the parallel projection in the direction of − e : π : conv(0 , e , e , e , e + e , e + e ) → conv(0 , e , e )We have π = f Γ for Γ = { [0 , e ] } . However, π is not a deflation because π ( e + e )and π ( e + e ) are neither vertices not interior points of the target polytope. Example 5.3 ( Not all surjective vertex maps are face-collapses ) . By a slight mod-ification of the map above, we get an example of a surjective vertex map which isnot a face-collapse. Consider the parallel projection in the direction − e : ρ : conv(0 , e , e , e + e , e + e ) → conv(0 , e , e )This projection cannot be a face-collapse because none of the positive dimensionalfaces of the source polytope is collapsed into a point. To show that ρ is a vertex,assume there is an affine 1-family ( ρ t ) ( − , with ρ = ρ . Then it must be constant onthe vertices of conv(0 , e , e ). But it must also be constant on e + e and e + e . Infact, if the family is not constant there then ρ t ( e + e ) and ρ t ( e + e ) must trace out OM-POLYTOPES 19 e e e e e ρ Figure 5.
A surjective vertex map that is not a face-collapseparallel intervals as t varies over ( − ,
1) (one uses barycentric coordinates). But, onthe other hand, these trajectories must be confined to the non-parallel edges [0 , e ]and [0 , e ] of the target polytope. So the family ( ρ t ) ( − , is constant on all verticesof the source polytope, forcing the family to be constant – a contradiction. Example 5.4 ( Many incident facets ) . As we know, dim(Hom( P n , P n )) = 6 for theregular n -gon, n ∈ N . By Proposition 2.1(1), any automorphism P n → P n sits on 2 n facets of Hom( P n , P n ). These 2 n facets can be split into two groups of n facets, eachdefining an edge of Hom( P n , P n ). For the identity map, these groups are specifiedas follows. F = (cid:8) H (cid:0) ι ( ζ kn )) , [ ζ kn , ζ k +1 n ] (cid:1) ⊂ Hom( P n , P n ) | k ∈ Z (cid:9) , F = (cid:8) H (cid:0) ι ( ζ kn )) , [ ζ k − n , ζ kn ] (cid:1) ⊂ Hom( P n , P n ) | k ∈ Z (cid:9) , notation as in Proposition 2.1(1). The edges are, correspondingly, E = \ F F = (cid:26) ι t : P n → P n , ζ kn (cid:18) − t (cid:19) ζ kn + (cid:18)
12 + t (cid:19) ζ k +1 n (cid:27) t ∈ [ − , , and E = \ F F = (cid:26) ρ t : P n → P n , ζ kn (cid:18) − t (cid:19) ζ k − n + (cid:18)
12 + t (cid:19) ζ kn (cid:27) t ∈ [ − , . We leave to the reader to check that E and E are in fact the edges, joining theidentity map with the rotations by 2 π/n and − π/n , respectively. Example 5.5 ( Gaps in ranks ) . For two polytopes P and Q , the range of possibleranks r of the vertices of Hom( P, Q ) is0 ≤ r ≤ min(dim P, dim Q ) . For any polytopes P and Q , the polytope Hom( P, Q ) always has rank 0 vertices. Ifdim P, dim Q >
Claim.
For any natural number n there are no rank 2 vertices in Hom( ♦ n , ∆ ). Figure 6.
Two edges of Hom( P n , P n ) incident to the identity map, n = 6First, for every polytope P and an element f ∈ vert( ♦ n , P ), one hasvert(Im f ) ⊂ vert( P ∩ ( − P ( c ) )) , where c is the center of Im f and − P ( c ) is the symmetric image of P w.r.t. c . Infact, because Im f is centrally symmetric, we have Im f ⊂ P ∩ ( − P ( c ) ). But ifthere is a vertex v ∈ Im f not in vert( P ∩ ( − P ( c ) )), then there is an open interval I ⊂ Im f , containing v . Then, by sliding v along I (and − v ( c ) along − I ( c ) in theopposite direction), one can define an affine 1-parameter ( f t ) ( − , ⊂ Hom( ♦ n , P )with f = f . Such is not possible in view of (pc ).Returning to the case P = ∆ , assume to the contrary that f is a rank 2 vertex ofHom( ♦ n , ∆ ). Let c be the center of Im f . There are two cases: (i) ∆ ∩ ( − ∆ ( c )2 ) isa parallelogram, or (ii) ∆ ∩ ( − ∆ ( c )2 ) is a centrally symmetric hexagon. In the firstcase, there is a vertex x ∈ ∆ such that − x ( c ) is in the interior of the edge E ⊂ ∆ ,opposite to x . So sliding c along a small open interval c ∈ I , parallel to E , producesan affine family of parallelograms (∆ ∩ ( − ∆ ( c )2 )) I in ∆ . The latter can be used todefine an affine 1-parameter family ( f t ) ( − , ⊂ Hom( ♦ n , ∆ ) with f = f . In thesecond case, we use the similar sliding procedure, except now the small open interval I ⊂ ∆ , containing c , is not constrained to have any particular direction – we canalways define an affine family of centrally symmetric hexagons (∆ ∩ ( − ∆ ( c )2 )) I . Ineither case we get a contradiction by (pc ).As a side observation, the right picture in Figure 7 gives rise to yet anotherexample of a deflation, followed by an injective vertex map, such that the compositeis not a vertex map. In fact, the deflation is a surjective affine map ♦ → ∆ ∩ ∆ ( c ) c and the injective vertex map is the embedding ∆ ∩ ∆ ( c ) c → ∆ . One needs to applya variant of the sliding argument in Example 5.1 to show that the latter belongs tovert (cid:0) ∆ ∩ ∆ ( c ) c , ∆ (cid:1) . The crucial point is that no perturbation of the hexagon in∆ can keep both the isomorphism class of the hexagon and the ‘hexagon vertex ↔ triangle edge’ incidence table constant.6. Generic pairs of polygons
The goal of this section is to understand some of the structure of the hom-polytopeof a generic pair of polygons (
P, Q ). The main result, Theorem 6.1, is that such a
OM-POLYTOPES 21 c ∆ -∆ ( c )2 c ∆ -∆ ( c )2 Figure 7.
The sliding argument in Example 5.5hom-polytope is ’almost simple’: apart from the short list of vertex maps of rankzero and one, every vertex of Hom(
P, Q ) is simple.We begin by constructing spaces of polytopes that will allow us to precisely statethis result. Fix dimensions d and e and integers m ≥ d + 1, n ≥ e + 1. To everyreal d × m matrix M we associate the polytope P ( M ) ⊆ R d given as the convexhull of the columns of M . Let R d,m ⊆ R d × m be the set of matrices M for which thecolumns are in convex position and affinely span R m ; that is, for which P ( M ) is a d -polytope with m vertices. Similarly, to every real n × e matrix M ′ we associatethe polyhedron Q ( M ′ ) ⊆ R e given by the system of inequalities M ′ x ≤ , where ∈ R n denotes the vector of all ones. Let R ′ e,n be the set of matrices M ′ for which Q ( M ′ ) is an e -polytope with n facets.Note that the conditions defining R d,m and R ′ e,n are stable under small pertur-bation; that is, R d,m and R ′ e,n are open subsets of R d × m and R n × e , respectively.Also, for generic M and M ′ , P ( M ) is simplicial and Q ( M ′ ) is simple. Furthermore, R d,m ⊂ R d × m and R ′ e,n ⊂ R n × e are semi-algebraic subsets (i. e., defined by algebraicequalities and (strict) inequalities). In fact, the convex d -polytopes with m -verticesgive rise to only finitely many combinatorial types and each type is represented by asemi-algebraic subset of R d × m , consisting of the matrices whose certain d × d minorsvanish, certain d × d minors are positive, and certain d × d minors are negative.Therefore, R d,m is the union of finitely many semi-algebraic sets and, as such, isitself semi-algebraic. A similar argument applies to R ′ e,n .Clearly, R ,m and R ′ ,n are the realization spaces of single combinatorial types.Moreover, the realization space of any 3-dimensional combinatorial type is a clas-sically a smooth ball; however, starting from d = e = 4, the realization space of acombinatorial type can be arbitrarily complicated; see [10].It is in the context above that we consider the hom-polytopes Hom( P ( M ) , Q ( M ′ ))for a generic pair ( M, M ′ ).We now focus on the case of pairs of polygons: d = e = 2. Every m -gon inthe plane is associated to a matrix in R ,m that is unique up to permuting itscolumns. Every n -gon in the plane that contains the origin in its interior is associated to a matrix in R ′ ,n that is unique up to permuting its rows. So R ,m × R ′ ,n , afull-dimensional semi-algebraic open subset of the Euclidean space R × m × R n × ,effectively represents the space of pairs of polygons.One the one hand, we have dim( R ,m × R ′ ,n ) = 2 m + 2 n . On the other hand, forevery pair ( M, M ′ ) ∈ dim( R ,m × R ′ ,n ), the space of all small perturbations of thefacets of Hom( P ( M ) , Q ( M ′ )), keeping the facet-normals invariant, has dimension mn (by Proposition 2.1(1)). Yet, we have Theorem 6.1.
There is a dense open subset U m,n of R ,m × R ′ ,n such that if ( M, M ′ ) ∈ U m,n , then every vertex map f : P ( M ) → Q ( M ′ ) of rank two is asimple vertex of Hom( P ( M ) , Q ( M ′ )) .Proof. Write P := P ( M ), Q := Q ( M ′ ). Since d = e = 2, Hom( P, Q ) is a six-dimensional polytope and we can write M = (cid:18) s . . . s m t . . . t m (cid:19) , M ′ = u v ... ... u n v n . To verify the conclusion of Theorem 6.1 for fixed P and Q , it suffices to show thestronger statement that if a map f ∈ aff( R , R ) lies in the intersection of the affinehulls of seven facets of Hom( P, Q ), then either • f ( P ) is not contained in Q , or • f is not of full rank.For each i = 1 , . . . , n and j = 1 , . . . , m there is a facet of Hom( P, Q ) whose affinespan is given by the equation ( u i , v i ) · f ( s j , t j ) = 1 . If we write f ( x, y ) = ( αx + βy + γ, δx + εy + ζ ), then more explicitly, this affinespan is given by (cid:0) u i v i (cid:1) α β γδ ε ζ s j t j = 1 , or as a linear constraint on the entries of f ,( u i s j ) α + ( u i t j ) β + u i γ + ( v i s j ) δ + ( v i t j ) ε + v i ζ = 1 . In other words, the condition for the affine hulls of seven facets to meet is that thelast column in the matrix below is a linear combination of the first six columns: A i ,...,i ,j ,...,j = u i s j u i t j u i v i s j v i t j v i − u i s j u i t j u i v i s j v i t j v i − u i s j u i t j u i v i s j v i t j v i − ∈ R × So the matrix A i ,...,i ,j ,...,j needs to be singular for some distinct pairs of indices( i , j ) . . . , ( i , j ). Note that the indices i , . . . i need not all be distinct, nor do j , . . . j .Thus it will suffice to show that for all choices of ( i , j ) , . . . , ( i , j ), either OM-POLYTOPES 23 • the seven facets do not intersect in any point of Hom( P, Q ), or • every map in the intersection of the seven facets is of less than full rank, or • the generic matrix A i ,...,i ,j ,...,j is nonsingular.We index the various cases for lists of indices by coincidence graphs : bipartitegraphs G = ( A, B, E ) with exactly seven edges and no isolated nodes. The set A represents the distinct elements of the list i , . . . i (facets of Q ) and B representsthe distinct elements of j , . . . , j (vertices of P .) The nodes of G are not labelledbut the two parts A and B of G are distinguishable: the condition that two verticesof P land on the same edge line (i.e. affine span of an edge) of Q is not the same asthe condition that a vertex of P lands on the intersection of two different edge linesof Q . For instance, the case that i , . . . , i are all distinct and j , . . . , j are also alldistinct is encoded by the graph with seven vertex-disjoint edges, the first graph inTable 1. The case that j = j , i = i , and everything else is distinct is encoded bythe third graph in the left column of Table 1. Observation 6.2. (1) If A contains a node of degree greater than two, then there is a vertex v of P such that f ( v ) is a point of intersection of three distinct edge lines of Q .But no three edge lines of a polygon can meet at a point (whether this pointis inside or outside of the polygon).(2) If B contains a node of degree greater than two, then f sends three differentvertices of P onto the same edge line of Q . Since any three vertices of P areaffinely independent, this implies that f sends all of R onto the same line;i.e, f is not of full rank.(3) If G contains a 4-cycle, then f sends two vertices of P to the same point:the intersection of two edge lines of Q . So again f is not of full rank.(4) If G contains a 6-cycle, then there are vertices v , v , v of P and edge lines ℓ , ℓ , ℓ of Q such that f ( v ) = ℓ ∩ ℓ , f ( v ) = ℓ ∩ ℓ , and f ( v ) = ℓ ∩ ℓ .For the condition f ( P ) ⊆ Q to also be satisfied, the three intersection points ℓ ∩ ℓ , ℓ ∩ ℓ , and ℓ ∩ ℓ must all be vertices of Q . That is, Q is a triangle.Then since f ( P ) contains all three vertices of Q , f ( P ) = Q , which impliesthat P is also a triangle. Then Hom( P, Q ) = Q , the product of threetriangles, and then all of its vertices are simple.In summary, we may now assume that G has no cycles and no vertices of degreegreater than two. That is, G is a union of vertex-disjoint paths with exactly sevenedges. The 31 such graphs (with distinguished upper part A and lower part B ) areshown in Table 1.For each graph G , we used the computer algebra system Macaulay 2 [6] to com-pute the appropriate generic determinant D G and to verify that it is not identi-cally zero. It follows that in each case, D G is nonzero on a dense open subset of R m × × R × n , the complement of an algebraic hypersurface. Since R ,m and R ′ ,n are full-dimensional subsets of R m × and R × n , we conclude that D G is also nonzeroon a dense open subset U G of R ,m × R ′ ,n . We now let U n,m := T G U G , where the Table 1.
The 31 possible coincidence graphsintersection is taken over the 31 coincidence graphs G . For every pair ( M, M ′ ) in U n,m , every full-rank vertex of Hom( P, Q ) is simple. (cid:3)
Corollary 6.3.
For every m, n ≥ there is a dense open subset V m,n of R ,m × R ′ ,n such that for n approaching infinity and fixed m , the ratio of simple vertices to allvertices in Hom(
P, Q ) tends to one under that condition ( M, M ′ ) ∈ V m,n .Proof. We take V m,n to be the dense open subset of U m,n given by the additionalconstraint that P has no parallel edges. For fixed ( M, M ′ ) ∈ V m,n , we group thevertex maps f ∈ Hom(
P, Q ) into four classes as follows.(1) f has rank zero; that is, f maps all of P to a single vertex of Q . Then f isnot a simple vertex unless m = 3.(2) f has rank one and im( f ) is an edge of Q . Again f is not simple unless m = 3.(3) f has rank one and im( f ) is a proper diagonal of Q . Then by Corollary 4.3(1), f is a face collapse; that is, it maps an edge of P to one end of the diagonaland (since P has no parallel edges) a vertex to the other end. All othervertices of P are mapped into the interior of Q . This means f satisfiesexactly six facet inequalities of Hom( P, Q ), so it is simple.(4) f has rank two. Then f is simple by Theorem 6.1. OM-POLYTOPES 25
By an easy refinement of Corollary 4.3(4), we see that there are 2 mn vertex mapsof type (2), and mn ( n −
2) of type (3). There are also n of type (1). Let k be thenumber of vertex maps of type (4), which may depend on the particular choice of( M, M ′ ) ∈ V m,n . Regardless of this choice, the ratio of simple to total vertex mapsis (if m >
3) at least mn ( n −
2) + kn + 2 mn + mn ( n −
2) + k ≥ mn ( n − n + mn = n − m − + n , which tends to one as n tends to infinity. (cid:3) We now return to the case of arbitrary dimension, where we make the followingconjecture which would generalize Theorem 6.1.
Conjecture 6.4.
For any dimensions d and e and any integers m ≥ d + 1 and n ≥ e + 1 , there is a dense open subset U d,e,m,n of R d,m × R ′ e,n such that if ( M, M ′ ) ∈ U d,e,m,n , then every vertex map f : P ( M ) → Q ( M ′ ) of full rank is a simple vertexof Hom(
P, Q ) . The motivation is that the main idea of the proof of Theorem 6.1, which isto use a finite list of graphs to enumerate the situations under which a vertex ofHom( P ( M ) , Q ( M ) might not be simple and then to show that each is equivalent tothe nonvanishing of a certain generic determinant, does not appear to depend on thedimensions of the source and target polytopes. However, some of the conditions inObservation 6.2 do not apply in the more general setting, and the number of graphsthat must be checked grows very rapidly with d and e .7. Regular polygons
We now consider the hom-polytopes of pairs ( P m , P n ) of regular polygons. Theredo not appear to be general descriptions of all vertices of Hom( P m , P n ), or even ofHom( P n , P n ), so we only consider special cases. Again the main goal is to understandthe set of vertices. In fact we need only consider those of full rank, since we havethe following immediate consequence of Corollary 4.3(4). Corollary 7.1.
The number of rank zero vertices of
Hom( P m , P n ) is n and thenumber of rank one vertices is (cid:26) mn ( n −
1) for m odd, mn ( n − for m even.For any positive integer k ≥
3, let Z k be the cyclic group of order k and let D k be the dihedral group of order 2 k . Then D k acts naturally on P k by rotation andreflection, with Z k identified with the subgroup of rotations inside D k . In particular,this induces an action of D m × D n on Hom( P m , P n ). The action respects faces ofeach dimension. The action of the subgroup Z m × Z n is free and transitive on the mn facets of Hom( P m , P n ). Although the action on the vertices is more subtle, it isstill useful in enumerating them. We begin with a sample result that holds for all m and n and serves as a mild check on the experimental data we will later present. Proposition 7.2.
Fix m and n and let V be the number of vertices of Hom( P m , P n ) .Then (1) n divides V , (2) m divides V − n , and (3) n and V have the same parity.Proof. All of these statements follow from applying the orbit-stabilizer theorem tothe action of various subgroups of D m × D n on vert(Hom( P m , P n )).(1) The action of any nontrivial rotation in Z n on Hom( P m , P n ) fixes only one affinemap: the zero map. The zero map is not a vertex of Hom( P m , P n ), so the action isfree on the vertices.(2) The action of any nontrivial rotation in Z m on Hom( P m , P n ) fixes only vertexmaps of rank zero. Thus m divides V − n .(3) If n is even, this is immediate from (1).If n is odd, then a reflection s ∈ D n fixes only the line segment from one vertexof P n to the midpoint of the opposite edge. Thus the only maps in Hom( P m , P n )that are fixed by s are the maps that take P m to that segment. But then, by theperturbation criterion (pc ), the only vertex of Hom( P m , P n )) that is fixed by s isthe map that takes all of P m to the mentioned vertex of P n . It follows that 2 divides V −
1, so V is odd. (cid:3) Proposition 7.3.
For any n ≥ , Hom( P , P n ) = P n . In particular, the number offull-rank vertices is n ( n − n − .Proof. The first statement is a special case of Corollary 3.6(1). For the second, notethat a map f is a full rank vertex of Hom( P , P n ) if and only if it sends each vertexof P to a distinct vertex of P n . (cid:3) Proposition 7.4.
The number of full rank vertices of
Hom( P n , P ) is (cid:26) n ( n +1)( n − for n odd n ( n − n − for n even Proof.
Suppose f is a rank two vertex of Hom( P, P ) for any polygon P . Then f cannot send more than two vertices to any edge of P . Since it lies at the intersectionof at least six different facets of Hom( P, P ), it must send exactly two vertices of P to each edge of P . That is, it inscribes an affine image of P into the equilateral P . But since all triangles in the plane are affinely isomorphic, this is equivalent tocircumscribing an arbitrary triangle around P itself.Now specialize to P = P n and let its edges be E , E , . . . , E n − in consecutiveorder. The condition that a triangle can be drawn around P n along three edges E i , E j , E k is that the gap between each pair of indices, taken cyclically, is at most ⌊ ( n − / ⌋ . We count such triples separately in the cases where n is odd and whereit is even.Suppose n = 2 p + 1 is odd. Consider just the triples 0 = i < j < k . Then wemust have 1 ≤ j ≤ p, k − j ≤ p, (2 p + 1) − k ≤ p. OM-POLYTOPES 27
That is, for j fixed we must have p + 1 ≤ k ≤ p + j , giving P pj =1 j = (cid:0) p +12 (cid:1) tripleswith i = 0. Since n of the allowed triples include 0, the total number of triples is n (cid:18) p + 12 (cid:19) = ( p + 1) p (2 p + 1)6= (2 p + 2)2 p (2 p + 1)24= ( n + 1)( n − n . Similarly if n = 2 p is even, the size of each gap must be at most p −
1. A similarcalculation gives the formula n ( n − n − for the number of triples of edges in thiscase.(Note that these formulae appear without proof as sequence A060422 in [11], listedas the number of triples of vertices of a regular n -gon that form acute triangles. Bypassing to the dual polygons with respect to 0, it is not hard to see that this isequivalent.)Finally, once we have chosen the three edges E i , E j , E k of P n , we can apply anarbitrary symmetry of P , so we multiply the formulae above by 3! = 6 to obtainthe total number of rank two vertices of Hom( P n , P ). (cid:3) Proposition 7.5.
The number of vertices of
Hom( P m , P ) is (cid:26) (2 m + 2) for m odd( m + 2) for m even. In particular the number of full-rank vertices is (cid:26) m − m for m odd m − m for m even Proof.
By Proposition 2.1(3), Hom( P m , P ) = Hom( P m , I ) × Hom( P m , I ). NowHom( P m , I ) has two vertices of rank zero and (by Corollary 4.3) has 2 m vertices ofrank one if m is odd or m such vertices if m is even. (cid:3) Note that if m is even, we obtain an explicit description of Hom( P m , I ) fromCorollary 3.6(2): it is a bipyramid over the dual m -gon P m . The f -vector (i. e.,the vector of the numbers of vertices, edges, 2-faces etc) of this bipyramid is ( m +2 , m, m, f i ( P × Q ) = i X j =0 f j ( P ) f i − j ( Q ) , we compute the f -vector of Hom( P m , I ) to be( m + 4 m + 4 , m + 12 m, m + 4 m, m + 2 m + 4 , m + 6 m, m, . Proposition 7.6.
The number of full-rank vertices of
Hom( P , P n ) is (cid:26) n − n for n odd n − n + 6 n for n even Proof.
Let v , v , v , v and w , w , . . . , w n − respectively be the vertices of P and of P n in counterclockwise order, with v = w . If f : P → P n is a vertex map of ranktwo, then in order to achieve the necessary six incidences of vertices of f ( P ) withfacets of P n , one of the following must apply:(1) f sends at least three vertices of P to distinct vertices of P n , or(2) f sends two adjacent vertices of P to distinct vertices of P n and the othertwo onto interior points of edges, or(3) f sends two opposite vertices of P to distinct vertices of P n and the othertwo onto interior points of edges.In case (1), we first observe that an affine map f : P → P n which maps threevertices of P to vertices of P n is automatically a vertex map; see the comment atthe beginning of Example 5.1. We may assume up to symmetry that f ( v ) = w , f ( v ) = w i , and f ( v ) = w n − j with 0 < i ≤ j . We now consider different cases forthe interior angle Θ of the parallelogram f ( P ) at w .If Θ is acute, then f ( v ) lies outside the unit circle. In particular it is not in P n ,so no vertex maps are obtained this way.If Θ is a right angle, then f ( v ) is a rectangle and we must have that n = 2 p iseven and i + j = p . Furthermore f ( v ) is also a vertex of P n : specifically it is thevertex w p directly opposite w , as in the first two pictures in Table 3. Thus to fixthe image of such a map, we must pick two opposite pairs of vertices of P n . Finally,taking into account the eight possible orientations of f ( P ), the number of thesemaps is 8 (cid:18) p (cid:19) = 4 p ( p −
1) = n ( n −
2) = n − n. Finally, if Θ is obtuse then f ( v ) is necessarily in the interior of P n (one uses thesymmetry w.r.t. to the perpendicular line through the midpoint of [ w , w n − j ]). Inparticular, f is a valid map from P to P n . The condition for Θ to be obtuse is that i + j < n/
2. This situation holds in the third and fourth pictures in Table 2 (with n = 7 and respectively i = j = 1; i = 1 , j = 2) and in the third and fourth picturesin Table 3 (with n = 8 and respectively i = j = 1; i = 1 , j = 2.)By considerations similar to the above, the number of ways to obtain such a mapis 8(2 p + 1) (cid:18) p (cid:19) = n ( n − n −
3) = n − n + 3 n if n = 2 p + 1, or8(2 p ) (cid:18) p − (cid:19) = 2 p (2 p − p −
4) = n ( n − n −
4) = n − n + 8 n if n = 2 p .In case (2), we may assume that f ( v ) = w and f ( v ) = w i for some i < n/ P n is symmetric w.r.t. the perpendicular line through the midpointof [ w , w i ], we must actually have that f ( P ) is a rectangle in order that both v and v land on edges. Furthermore n must be odd or f ( v ) and f ( v ) will both bevertices of P n , a situation we already considered in case (1). Finally, we must have OM-POLYTOPES 29
Table 2.
Rank two vertices of Hom( P , P ) up to symmetry Table 3.
Rank two vertices of Hom( P , P ) up to symmetry i + 1 < n/ f ( v ) and f ( v ) are contained in P n .This situation holds in the first and second pictures in Table 2, with n = 7 andrespectively i = 1; i = 2.Setting n = 2 p + 1, we see that the number of maps of this type is8(2 p + 1)( p −
1) = 4 n ( n −
3) = 4 n − n. That all these maps are in fact vertex maps follows from (pc ): any affine 1-familymust be constant on v and v , but if such a family is not constant on v thenthe images of v and v , when the family parameter varies over ( − , non-parallel edges of P n .In case (3), we may assume f ( v ) = w and f ( v ) = w i for some 0 < i ≤ n/ i = n/
2, then n is even. Then we can indeed arrange that f ( v ) and f ( v ) areboth on edges, but these will necessarily be opposite (and hence parallel) edgesof P n . Without moving v or v , we can then slide v and v along these edgesin the opposite directions to produce an affine 1-family of maps, contradicting theassumption that f is a vertex map.On the other hand, if i < n/
2, and we assume that f ( v ) is on an edge, let Q be the polygon with vertices w , w , . . . , w i and Q ′ be the polygon obtained byreflecting Q across the line defined by w and w i . Since f ( v ) = w , f ( v ) = w i , and f ( w ) ∈ ∂Q , we conclude that f ( w ) ∈ ∂Q ′ . But Q ′ touches the edges of P only at w and at w i , so we cannot arrange that f ( v ) lies on an edge of Q . That is, thereare no vertices of Hom( P , P m ) of this type: case (3) is impossible.Adding up the vertices described by the various cases, we obtain the total countof rank two vertices of Hom( P , P n ) as claimed in the proposition. (cid:3) m n rank 0 rank 1 rank 2 total3 3 3 18 6 273 4 4 36 24 643 5 5 60 60 1253 6 6 90 120 2163 7 7 126 210 3433 8 8 168 336 5124 3 3 12 0 154 4 4 24 8 364 5 5 40 80 1254 6 6 60 72 1384 7 7 84 280 3714 8 8 112 240 3605 3 3 30 30 635 4 4 60 80 1445 5 5 100
60 165
540 696
770 987 m n rank 0 rank 1 rank 2 total6 3 3 18 12 336 4 4 36 24 646 5 5 60
240 305
84 180
864 1040
770 915
700 1001
400 485
288 414
912 1144
Table 4.
Expected numbers of vertices of Hom( P m , P n )7.1. Experimental results.
We end with a table of experimental results for thenumber of vertices of Hom( P m , P n ) for all m, n ≤
8. Our approach to the com-putation was as follows. We begin with rational approximations Q m and Q n andcompute the polytope H := Hom( Q m , Q n ) with exact arithmetic, using the softwarepackage Polymake [5]. This is possible because of the explicit facet description ofany hom-polytope given by Proposition 2.1.However, we do not expect Hom( Q m , Q n ) to have the same number of vertices asHom( P m , P n ). For example, if m = n ≥
5, the affine map that rotates P m by πm is avertex of Hom( P m , P m ) at which 2 m facets meet; see Example 5.4. This map doesnot exist in Hom( Q m , Q m ). Specifically, the corresponding facets do not all meetin one point, but various subsets of them do meet to form several different verticesof Hom( Q m , Q m ). Our problem, then, is to identify the collections of vertices ofHom( Q m , Q n ) that correspond to single vertices of Hom( P m , P n ).Given a polytope R and ε >
0, we say that a collection V of vertices of R is an ε -cluster if k v − w k < ε for all v, w ∈ V . If all of the vertices of R are partitionedinto a collection of disjoint clusters, we say that ε successfully partitions vert( R ).Note that for any sufficiently large ε , we get a single cluster, and for any sufficientlysmall ε , each vertex forms a cluster by itself. However, we need intermediate valuesof ε that successfully partition vert(Hom( Q m , Q n )). By trying several values, we findthat for six-digit rational approximations Q m , Q n and for all m, n ≤
8, the values ε = 10 − and ε = 10 − give the same nontrivial partition. Furthermore, the data OM-POLYTOPES 31 resulting from such a partition agree with what we have proved for the cases m = 3, m = 4, n = 3, and n = 4, and also with Proposition 7.2.Using this partition, we predict the vertex counts shown in the slanted entries ofTable 4; all other values can be obtained from the theoretical results in this section. References [1]
Polymake , software for the algorithmic treatment of convex polyhedra, http://polymake.org/doku.php .[2] A. Akopyan and R. Kerasev. Inscribing a regular octahedron into polytopes.
Preprint ,http://arxiv.org/abs/1107.4428.[3] Louis J. Billera and Bernd Sturmfels. Fiber polytopes.
Ann. of Math. (2) , 135(3):527–549,1992.[4] Winfried Bruns and Joseph Gubeladze.
Polytopes, rings, and K -theory . Springer Monographsin Mathematics. Springer, Dordrecht, 2009.[5] Ewgenij Gawrilow and Michael Joswig. Polymake: a framework for analyzing convex poly-topes. In Polytopes—combinatorics and computation (Oberwolfach, 1997) , volume 29 of
DMVSem. , pages 43–73. Birkh¨auser, Basel, 2000.[6] Daniel R. Grayson and Mike Stillman. Macaulay 2, a software system for research in algebraicgeometry. Available from .[7] G. M. Kelly. Basic concepts of enriched category theory.
Repr. Theory Appl. Categ. , (10),2005. Reprint of the 1982 original [Cambridge Univ. Press, Cambridge].[8] Saunders Mac Lane.
Categories for the working mathematician , volume 5 of
Graduate Textsin Mathematics . Springer-Verlag, New York, second edition, 1998.[9] P. G. L. Porta Mana. Conjectures and questions in convex geometry (of in-terest for quantum theory and other physical statistical theories).
Preprint , http://arxiv.org/abs/1105.3238/ .[10] J¨urgen Richter-Gebert and G¨unter M. Ziegler. Realization spaces of 4-polytopes are universal. Bull. Amer. Math. Soc. (N.S.) , 32:403–412, 1995.[11] N.J.A Sloane. The on-line encyclopedia of integer sequences. published electronically athttp://oeis.org, 2008.[12] L. Valby. A category of polytopes. available at http://people.reed.edu/~davidp/homepage/students/valby.pdf .[13] G¨unter Ziegler.
Lectures on polytopes , volume 152 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1998, Revised edition.
Departamento de Matem´aticas, Universidad de los Andes, Cra 1 No. 18A-10,Edificio H, Bogot´a, 111711, Colombia
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